Assume an iteroparous population which occupies two different habitats.
New-borns disperse and settle in one of the two habitats where they stay for the rest of their life. Mutational change occurs in the habitat specific adult survival probabilitiest11andt22, which are assumed to be traded off. We dis-tinguish two scenarios of population regulation. (i) Adult fertility depends on one common resource (e.g., freely floating plankton) and therefore the offspring number decreases with increasing total population size N1+N2. (ii) Adult fertility depends on a local resource (e.g., space within each habi-tat) and therefore habitat specific fecundities decrease with local population densities. Invasion fitness is given by
w(θ0, θ) =f11Df11(θ) +t11(θ0) +f22Df22(θ) +t22(θ0)− (16) (f11Df11(θ) +t11(θ0))(f22Df22(θ) +t22(θ0)) +f12Df12(θ)f21Df21(θ).
First we consider the case with global competition. From section 3.2 we know that I = ˆN1+ ˆN2 is an optimisation criterion. An optimisation cri-terion ψ can be found by solving w(θ, I(θ)) = 1 for I(θ). However, both Nˆ1+ ˆN2 and ψ are lengthy expressions that do not allow for an analytical treatment. Nevertheless, we can make the following general statements. Since evolution affects the diagonal componentsa11 and a22 the IBs are concave.
Therefore, any singular point on a convex trade-off is necessarily a repeller.
Conversely, for a singular point to be a CSS the trade-off curve has to be more strongly concave than the IB. With symmetric parameter-values the habitat generalist, characterised byθ= 0.5, is a singular point. For this
gen-eralist the bifurcation from a repeller to a CSS has to occur for somez >1.
Figure 5a shows a numerically calculated bifurcation diagram confirming our qualitative predictions.
Next we analyse the case where fecundities are decreasing functions of lo-cal densities: Df11(N1), Df21(N1), Df22(N2), Df12(N2). In this case the feed-back environment is given by I = ( ˆN1,Nˆ2) and selection is frequency-dependent. Given symmetric parameter-values we can apply the results of section 4. Assume that adults have equal fecundity in both patches (f11+ f21 = f22 +f12) and that juveniles are equally likely to settle in either patch, hence: f11 = f12 = f21 = f22. Furthermore, we assume that the trade-off is symmetric, that is, t11max = t22max (cf. eq. 3) and that all ju-veniles are equally susceptible to crowding: Df11 = Df12 = Df21 = Df22. From these symmetries follows that the habitat generalist withθ∗ = 0.5 is a singular point. From section 4 we can conclude that a threshold zt > 1 exists such that for all z > zt we can find an r ∈ R >0 such that for all θ0, θ∈B= (0.5−r,0.5 +r) withθ≶θ0≶0.5 we find∆DDC(θ0, θ)>0. This means that the singular point at θ∗ = 0.5 is locally uninvadable for z > zt (table 2). Conversely, for z < ztthe singular pointθ∗= 0.5 is locally invad-able because we can find a neighbourhood ofθ∗ where∆DDC(θ0, θ)<0 for θ ≶ θ0 ≶ 0.5. In order to understand the convergence properties of θ∗ we have to investigate ∆F DC(θ0, θ). Under the given symmetry assumptions we can prove that a neighbourhood B of θ∗ exists such that for θ0, θ ∈ B we find θ ≶ θ0 ≶ 0.5 ⇒ ∆F DC(θ0, θ) > 0 for all values of z (Appendix A). From table 2 we conclude that for z > zt the generalistθ∗ is a CSS. If
z is slightly smaller thanzt, then∆DDC(θ0, θ) becomes negative, however,
∆DDC(θ0, θ) +∆F DC(θ0, θ) stays positive andθ∗ turns into an evolution-ary branching point. Whenz becomes small enough such that the negative
∆DDC(θ0, θ) overrules the positive∆F DC(θ0, θ) the singular trait-valueθ∗
turns into an evolutionary repeller. Figure 5b shows a numerically calculated bifurcation diagram of singular points that confirms our qualitative predic-tions concerning the habitat generalist.
6 Discussion
In this article we classify a family of simple life-history models with respect to criteria driving the evolution in two traits that are connected by a trade-off.
Our main tools are a sign-equivalent and algebraically simpler expression for invasion fitness, curvature properties of invasion boundaries, the dimension of the feedback environment and the decomposition of invasion fitness into a density-dependent and a frequency-dependent component.
The results we present are not primarily motivated by questions about the evolution of specific life-cycles but rather by a desire to understand the mechanisms that govern the evolutionary dynamics in a larger class of mod-els. Our aim is to formulate principles of a more general nature that are independent of a specific model and it is these principles that we consider the most valuable result of our work. For the presented class of models the following conclusions can be drawn: (i) Trade-offs between an off-diagonal and a diagonal matrix component as well as between the two traits within a single matrix components correspond to linear invasion boundaries. In these
cases all singular points on trade-offs parameterised by z < 1 are suscep-tible to invasion by nearby mutants while the opposite holds true for sin-gular points on trade-offs parameterised by z > 1. (ii) Trade-offs between two diagonal components of the projection matrixA correspond to concave IBs. As a result, trade-off curves parameterised by z ∈ (0, zt) with zt > 1 give rise to singular points where populations experience disruptive selection.
Populations with mean trait-values equal to the singular trait-value are sus-ceptible to invasion by mutants with both smaller and larger trait-values. In models with frequency dependence this can lead to disruptive selection and phenotypic diversification. (iii) Trade-offs between two off-diagonal compo-nents correspond to convex IBs. As a result, trade-off curves parameterised by z ∈ (0, zt) with zt < 1 give rise to singular points where populations experience disruptive selection. Trade-off curves parameterised by z > zt
give rise to singular points where populations are not invadable by nearby mutants and experience stabilising selection. Hence, for a wide range of z-values such trade-offs favour the evolution of intermediate phenotypes. (iv) Trade-offs between traits that are both necessary to pass through both i-states result in frequency-independent selection. This scenario applies when both off-diagonal components of the population projection matrix consist of only a single term, that is, a transition from one i-state to the other is ei-ther only possible in terms of tkl or in terms of fkl. Under this condition the two evolving traits occur in a single product in the fitness function and it is this product that is maximised by selection. From (iii) we see that the majority of trade-off curvatures leads to intermediate phenotypes that strike
a balance between the conflicting traits. (v) Trade-offs between traits that are not both necessary to pass through both i-states are a prerequisite for frequency-dependent selection. Such traits affect different summands of the fitness function. Selection becomes frequency-dependent when each evolving trait occurs in a product with a function of density such that the traits are affected by differently weighted sums of the total population sizec1N1+c2N2. In the extreme case one evolving trait decreases with an increasing number of individual in i-state one while the other decreases with increasing N2. Such trade-offs give rise to either linear IBs (in case of a diagonal and an off-diagonal component or in case of two traits that affect a single matrix component) or concave IBs (in case of two diagonal components). From (i) and (ii) we see that in this case either all convex trade-off or all convex plus weakly concave trade-offs give rise to disruptive selection, facilitating the occurrence of evolutionary branching points.